Abstract
In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the formfor sufficiently small. Here , for , are smooth functions being an open bounded set. Then we use this result to compute the bifurcation functions which allow us to study the periodic solutions of the following T-periodic smooth differential systemIt is assumed that the unperturbed differential system has a sub-manifold of periodic solutions , . We also study the case when the bifurcation functions have a continuum of zeros. Finally we provide the explicit expressions of the bifurcation functions up to order 5.
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